Phân tích đa thức sau thành nhân tử: (Test)

 DẠNG TOÁN PHÂN TÍCH ĐA THỨC THÀNH NHÂN TỬ

1A = $- 6x + 9 - y^2 + x^2$

1. Rewrite it in the form $a^2−2ab+b^2$, where a=3 and b=x.
  $3^2−2(3)(x)+x^2−y^2$
2. Use Square of Difference:$(a−b)^2=a^2−2ab+b^2$.
  $(3−x)^2−y^2$
3. Use Difference of Squares: $a^2−b^2=(a+b)(a−b)$.
  $(3−x+y)(3−x−y)$

1B = $9x^2 — 16y^2$

1. Find the Greatest Common Factor (GCF).
  GCF = 33
2. Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
  $3(\Large{\frac{9x^2}{3}+\frac{−6y^2}{3}})$
3. Simplify each term in parentheses.
  $3(3x^2−2y^2)$

2A = $x^2 + 4x + 4 - y^2$

1. Rewrite it in the form $a^2+2ab+b^2$, where a=x and b=2.
  $x^2+2(x)(2)+2^2−y^2$
2. Use Square of Sum: $(a+b)^2=a^2+2ab+b^2$.
  $(x+2)^2−y^2$
3. Use Difference of Squares: $a2−b2=(a+b)(a−b)$.
  $(x+2+y)(x+2−y)$

2B = $25x^2 — 4y^2$

1. Rewrite it in the form $a^2−b^2$, where a=5x and b=2y.
  $(5x)^2−(2y)^2$
2. Use Difference of Squares: $a^2−b^2=(a+b)(a−b)$.
  $(5x+2y)(5x−2y)$

3A = $x^2 — 5xy - 4x + 20y$

1. Factor out common terms in the first two terms, then in the last two terms.
  x(x−5y)−4(x−5y)
2. Factor out the common term x−5y.
  (x−5y)(x−4)

3B = $4x^2 — 4x + 1 — y^2$

L = $4x^2 — 4x + 1 — y%2$
= $(4x^2 — 4x + 1) — y^2$
= $(2x)^2 -2.2x.1 +1^2 - y^2$   (có dạng $a^2 -2ab + b^2 = (a – b)^2$
= $[(2x)^2 -2.2x.1 +1^2] - y^2$
= $(2x – 1)^2 – y^2$     (có dạng $a^2 – b^2 = (a + b)(a - b)$
= $[(2x -1)+(y)] [(2x -1)-(y)]$
= $(2x -1 + y)(2x -1 –y)$

4A = $x^2 - 2xy + y^2 – 49$

1. Rewrite it in the form $a^2−2ab+b^2$, where a = x and b = y.
  $x^2−2(x)(y)+y^2−49$
2. Use Square of Difference: $(a−b)^2=a^2−2ab+b^2$.
  $(x−y)^2−49$
3. Rewrite it in the form $a^2−b^2$, where a=x−y and b=7.
  $(x−y)^2−7^2$
4. Use Difference of Squares: $a^2−b^2=(a+b)(a−b)$.
  $(x−y+7)(x−y−7)$

4B = $16x^2 - 4y^2$

1. Find the Greatest Common Factor (GCF).
  GCF = 4
2. Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
  $4\Large{(\frac{16x^2}{4}+\frac{−4y^2}{4})}$
3. Simplify each term in parentheses.
  $4(4x^2−y^2)$
4.Rewrite $4x^2−y^2$ in the form $a^2−b^2$, where a = 2x and b = y.
  $4((2x)^2−y^2)$
5. Use Difference of Squares: $a^2−b^2=(a+b)(a−b)$.
  $4(2x+y)(2x−y)$

5A = $x^2 — 12x + 36 — 81y^2$

1. Rewrite it in the form $a^2−2ab+b^2$, where a=x and b=6.
  $2x^2−2(x)(6)+6^2−81y^2$
2. Use Square of Difference: $2(a−b)^2=a^2−2ab+b^2$.
  $(x−6)^2−81y^2$
3. Rewrite it in the form $a^2−b^2$, where a=x−6 and b=9y.
  $(x−6)^2−(9y)^2$
4. Use Difference of Squares: $a^2−b^2=(a+b)(a−b)$.
  $(x−6+9y)(x−6−9y)$

5B = $9x^2 — y^2$

1. Rewrite it in the form $a^2−b^2$, where a=3x and b=y.
  $(3x)^2−y^2$
2. Use Difference of Squares: $a^2−b^2=(a+b)(a−b)$.
  $(3x+y)(3x−y)$

6A = $x^2 — 10xy + 25y^2 – 16$

1. Rewrite it in the form $a^2−2ab+b^2$, where a=x and b=5y.
  $x^2−2(x)(5y)+(5y)^2−16$
2. Use Square of Difference: $(a−b)^2 = a^2 − 2ab + b^2$.
  $(x−5y)^2−16$
3. Rewrite it in the form $a^2 − b^2$, where a = x − 5y and b = 4 .
  $(x − 5y)^2 − 4^2$
4. Use Difference of Squares:$ a^2 − b^2=(a + b)(a − b)$.
  $(x − 5y + 4)(x − 5y − 4)$

6B = $4x^2 - 36y^2$

1. Find the Greatest Common Factor (GCF).
GCF = 4
2. Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
  $4\Large{(\frac{4x^2}{4}+\frac{−36y^2}{4})}$
3. Simplify each term in parentheses.
  $4(x^2−9y^2)$
4.Rewrite $x2−9y2$ in the form $a2−b2$, where a=x and b=3y.
  $4(x^2−(3y)^2)$
5. Use Difference of Squares: $a^2−b^2=(a+b)(a−b)$.
  $4(x+3y)(x−3y)$

7A = $x^2 — 2x + 1 — y^2$

1. Rewrite it in the form $a2−2ab+b2$, where a=x and b=1.
  $x^2−2(x)(1)+1^2−y^2$
2. Use Square of Difference: $(a−b)^2=a^2−2ab+b^2$.
  $(x−1)^2−y^2$
3. Use Difference of Squares: $a^2−b^2=(a+b)(a−b)$.
  $(x−1+y)(x−1−y)$

8A = $49x^2 — 9y^2$

1. Rewrite it in the form $a^2−b%2$, where a=7x and b=3y.
  $(7x)^2−(3y)^2$
2. Use Difference of Squares: $a^2−b^2=(a+b)(a−b)$.
  $(7x+3y)(7x−3y)$

8A = $3x(5x + y) + 25x^2 — y^2$

  $3x(5x + y) + (5x)^2 — y^2$
  $3x(5x + y) + (5x - y)(5x +y)$
  $(5x + y) [3x +(5x - y)]$
  $(5x + y) (3x +5x - y)$
  $(5x + y) (8x - y)$

8B = $121x^2— 64y^2$

1. Rewrite it in the form $a^2 − b^2$, where a = 11x and b = 8y.
  $(11x)^2−(8y)^2$
2. Use Difference of Squares: $a^2 − b^2=(a + b)(a − b)$.
  $(11x + 8y)(11x − 8y)$

9A = $x^2 - 8x + 16 – y2$

1. Rewrite it in the form $a^2 − 2ab + b^2$, where a = x and b = 4.
    $x^2 − 2(x)(4) + 4^2 − y^2$
2. Use Square of Difference: $(a − b)^2 = a^2 − 2ab + b^2$.
  $(x−4)^2 − y^2$
3. Use Difference of Squares: $a^2 − b^2 = (a + b)(a − b)$.
  $(x – 4 + y)(x – 4 − y)$

11A = $x^2— 2x — y^2 + 1$

= $x^2 - 2x + 1 - y^2$
  = $(x^2 - 2x + 1) - y^2$
  = $(x – 1)^2 - y^2$
  = $(x – 1 + y)(x – 1 - y)$

11B = $16x^2 — 144y^2$

1. Find the Greatest Common Factor (GCF).
  GCF = 16
2. Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
  $16(x^2) – 16(9y^2)$
3. Simplify each term in parentheses.
  $16(x^2 − 9y^2)$
4.Rewrite $x2 − 9y2$ in the form $a^2 – b^2$, where a = x and b = 3y.
  $16(x^2 − (3y)^2)$
5. Use Difference of Squares: $a^2 – b^2 = (a + b)(a − b)$.
  $16(x + 3y)(x − 3y)$

12A = $x^2 — 6x + 9 — y^2$

1. Rewrite it in the form $a^2−2ab+b^2$, where a = x and b = 3.
  $x^2 − 2(x)(3) + 3^2 − y^2$
2. Use Square of Difference: $(a − b)^2 = a^2 − 2ab + b^2$.
  $(x−3)^2 − y^2$
3. Use Difference of Squares: $a^2 − b^2 = (a + b)(a − b)$.
  $(x – 3 + y)(x – 3 − y)$

12B = $81x^2 — 169ỵ^2$

1. Rewrite it in the form $a^2 − b^2$, where a = 9x and b = 13y.
  $(9x)^2 − (13y)^2$
2. Use Difference of Squares: $a^2 − b^2=(a + b)(a − b)$.
  $(9x + 13y)(9x − 13y)$

12C = $xy — 3x + y^2 - 3y$

  = $(xy + y^2) — (3x + 3y)$
  = $y(x + y) — 3(x +y)$
  = $(x + y) (y— 3)$

13A = $x^2 + 10x + 25 — 64y^2$

  = $x^2 + 10x + 5^2 — 64y^2$
  = $(x^2 + 10x + 5^2) — 8^2y^2$
  = $(x + 5)^2 — (8y)^2$
  = $[(x + 5) + 8y] [(x + 5) - 8y] $
  = $(x + 5 + 8y) (x + 5 - 8y) $

13B = $x^2 — 16y^2$

1. Rewrite it in the form $a^2 − b^2$, where a = x and b = 4y.
  $x^2 − (4y)^2$
2. Use Difference of Squares: $a^2 − b^2 = (a + b)(a − b)$.
  $(x + 4y)(x − 4y)$

C = $x^5 + x - 2x^4 – 2$

1. Factor out common terms in the first two terms, then in the last two terms.
  $x(x^4+1)−2(x^4+1)$
2. Factor out the common term $x^4+1$.
  $(x^4+1)(x−2)$

14A = $5x(x — y) + 9x — 9y$

  $5x(x — y) + 9(x — y)$
  = $(x — y)(5x + 9)$

14B = $196x^2 — 9y^2$

  = $(14x)^2 — (3y)^2$
  = $(14x)^2 — (3y)^2$
  = $(14x+3y)(14x−3y)$

15A = $x^2 — 4xy + 4y^2 — 36$

1. Rewrite it in the form $a^2 − 2ab + b^2$, where a = x and b = 2y.
  $x^2 − 2(x)(2y) + (2y)^2 − 36$
2. Use Square of Difference:   $(a − b)^2 = a^2 − 2ab + b^2$.
  $(x − 2y)^2 − 36$
3. Rewrite it in the form $a^2 − b^2$, where a = x − 2y and b = 6.
  $(x − 2y)^2 − 6^2$
4. Use Difference of Squares:$a^2 − b^2 = (a + b)(a − b)$.
  $(x − 2y + 6)(x − 2y − 6)$

15B = $225x^2 — 9y^2$

1. Find the Greatest Common Factor (GCF).
  GCF = 9
2. Factor out the GCF. (Write the GCF first. Then, in parentheses, divide each term by the GCF.)
  $9.25x^2 − 9.y^2)$
  $9\Large{(\frac{225x^2}{9} + \frac{−9y^2}{9})}$
3. Simplify each term in parentheses.
  $9(25x^2 − y^2)$
4. Rewrite $25x^2 − y^2$ in the form $a2 − b2$, where a = 5x and b = y.
  $9((5x)^2 − y^2)$
5. Use Difference of Squares:$a^2 – b^2$ = (a + b)(a − b).
  $9(5x + y)(5x − y)$

Bài 5a

Bài 55

Bài 6a

Bài 66

Bài 7a

Bài 77

Bài 8a

Bài 88

Bài 9a

Bài 99

Bài 10a

Bài 10

D = $ x^2 – 3$

  $a = x; b = \sqrt{3}$
  -> $E = x^2 – (\sqrt{3})^2$
E có dạng $a^2 – b^2 = (a +b)(a –b)$E = $(x + \sqrt{3})(x - \sqrt{3})$

E = $ x^3 – 2$

  2 = $(\sqrt[3]{2})^3$
E =$ x^3 - (\sqrt[3]{2})^3$
  a = x; b = $\sqrt[3]{2}$
  $a^3 – b^3 = (a – b)(a^2 + ab+ b^2)$
E =$(x - \sqrt[3]{2})(x^2 + \sqrt[3]{2} x+ (\sqrt[3]{2})2)$
  $(\sqrt[3]{2})^2 = 2^\frac{2}{3} =\sqrt[3]{2^2} $
E = $(x - \sqrt[3]{2}) (x^2 +\sqrt[3]{2} + \sqrt[3]{2^2} )$

F = $x^5 – 3x^3 – 2x^2 + 6$

1. Factor out common terms in the first two terms, then in the last two terms.
  $x^3(x^2−3)−2(x^2−3)$
2. Factor out the common term $x^2−3$
  $(x^2−3)(x^3−2)$

G = $x^2 - 2x + 1 - y^2$

1. Rewrite it in the form $a^2−2ab+b^2$, where a=x and b=1.
  $2x^2−2(x)(1)+1^2−y^2$
2. Use Square of Difference: $(a−b)^2=a^2−2ab+b^2$.
  $(x−1)^2−y^2$
3. Use Difference of Squares: $a^2−b^2=(a+b)(a−b)$.
  $(x−1+y)(x−1−y)$